In hydrogen atom, if `lamda_(1),lamda_(2),lamda_(3)` are shortest wavelengths in Lyman, Balmer and Paschen series respectively then `lamda_(1):lamda_(2):lamda_(3)` equals

Options

A. `1:4:9`
B. `9:4:1`
C. `1:2:3`
D. `3:2:1`

Explanation

To find the ratio of the shortest wavelengths in the Lyman, Balmer, and Paschen series of the hydrogen atom, we will use the formula for the wavelengths of spectral lines derived from the Bohr model of the hydrogen atom.

### Step-by-step Solution:

1. **Identify the Series**:
- The Lyman series corresponds to transitions to the n=1 level.
- The Balmer series corresponds to transitions to the n=2 level.
- The Paschen series corresponds to transitions to the n=3 level.

2. **Formulas for Wavelengths**:
The formula for the wavelength of light emitted during a transition from a higher energy level (n2) to a lower energy level (n1) is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant.

3. **Calculate \( \lambda_1 \) (Lyman Series)**:
- For the shortest wavelength in the Lyman series, \( n_1 = 1 \) and \( n_2 \) approaches infinity.
\[ \frac{1}{\lambda_1} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]
\[ \lambda_1 = \frac{1}{R} \]

4. **Calculate \( \lambda_2 \) (Balmer Series)**:
- For the shortest wavelength in the Balmer series, \( n_1 = 2 \) and \( n_2 \) approaches infinity.
\[ \frac{1}{\lambda_2} = R \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R \left( \frac{1}{4} - 0 \right) = \frac{R}{4} \]
\[ \lambda_2 = \frac{4}{R} \]

5. **Calculate \( \lambda_3 \) (Paschen Series)**:
- For the shortest wavelength in the Paschen series, \( n_1 = 3 \) and \( n_2 \) approaches infinity.
\[ \frac{1}{\lambda_3} = R \left( \frac{1}{3^2} - \frac{1}{\infty^2} \right) = R \left( \frac{1}{9} - 0 \right) = \frac{R}{9} \]
\[ \lambda_3 = \frac{9}{R} \]

6. **Find the Ratio \( \lambda_1 : \lambda_2 : \lambda_3 \)**:
- Now substituting the values we calculated:
\[ \lambda_1 = \frac{1}{R}, \quad \lambda_2 = \frac{4}{R}, \quad \lambda_3 = \frac{9}{R} \]
- The ratio is:
\[ \lambda_1 : \lambda_2 : \lambda_3 = \frac{1}{R} : \frac{4}{R} : \frac{9}{R} = 1 : 4 : 9 \]

### Final Answer:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 1 : 4 : 9 \]